Spectral numerical weather prediction models

503 Pages·2012·3.813 MB·English

by  Martin Ehrendorfer

Spectral Numerical
Weather Prediction
Models
OT124_Ehrendorfer_FM-10-14-11.indd 1 11/2/2011 11:31:40 AM
Spectral Numerical
Weather Prediction
Models
M E
artin hrEndorfEr
Innsbruck, Austria
Society for Industrial and Applied Mathematics
Philadelphia
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Copyright © 2012 by the Society for Industrial and Applied Mathematics
10 9 8 7 6 5 4 3 2 1
All rights reserved. Printed in the United States of America. No part of this book may
be reproduced, stored, or transmitted in any manner without the written permission of the
publisher. For information, write to the Society for Industrial and Applied Mathematics,
3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA.
Trademarked names may be used in this book without the inclusion of a trademark
symbol. These names are used in an editorial context only; no infringement of trademark
is intended.
IDL is a trademark of ITT Corporation.
LAPACK is a software package provided by Univ. of Tennessee; Univ. of California,
Berkeley; Univ. of Colorado, Denver; and NAG Ltd.
MacBook is a trademark of Apple, Inc., registered in the U.S. and other countries.
MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product
information, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA
01760-2098 USA, 508-647-7000, Fax: [phone], [email protected],
www.mathworks.com.
Subroutines four1, gauleg, gaussj, and plgndr2 in Section C.3 Copyright © 1986, 1992
Numerical Recipes Software. Used with permission.
Pictured on the cover is the prominent peak of the Vordere Brandjochspitze covered
by early winter snow (photo taken by the author November 27, 2010). As part of the
Karwendel ridge, the mountain’s peak towers at 2559 meters, north of the city of
Innsbruck. It fittingly provides the cover art for this book, as part of it describes the
details of the spectral numerical weather prediction model, PEAK. While flow simulation
over such complex mountain terrain remains a challenging subject, the yellow contours
display, on the basis of the simplest geophysical model, the roll-up stage in sheared
vorticity-conserving flow, simulated using the barotropic PEAK configuration.
Library of Congress Cataloging-in-Publication Data
Ehrendorfer, Martin.
Spectral numerical weather prediction models / Martin Ehrendorfer.
p. cm. -- (Other titles in applied mathematics)
Includes bibliographical references and index.
ISBN 978-1-611971-98-9 (pbk.)
1. Numerical weather forecasting--Mathematical models. I. Title.
QC996.E37 2011
551.63’4--dc23 2011028309

is a registered trademark.
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For Andreas, Bethany, Caroline,
Jakob, Johannes, Magdalena,
Matthias, Miriam, and Rebecca.
May they carry on.
r
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Contents
Preface xi
Acknowledgments xvii
List of Figures xxi
List of Tables xxiii
List of Algorithms xxv
I Atmospheric Dynamical Models 1
1 Introduction 3
1.1 Introduction—TheSpectral Method . . . . . . . . . . . . . . . . . . . . 3
1.2 A History of Numerical WeatherPrediction . . . . . . . . . . . . . . . 9
2 Governing Atmospheric Dynamics 13
2.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Global Atmospheric Energetics. . . . . . . . . . . . . . . . . . . . . . . 14
2.3 The Complete Governing Equations . . . . . . . . . . . . . . . . . . . . 19
2.4 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Potential Vorticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 The Primitive Equations 45
3.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 The Hydrostatic Primitive Equations . . . . . . . . . . . . . . . . . . . 46
3.3 Potential Vorticity in theHPEs . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5 The HPEs with Vertical Coordinate σ . . . . . . . . . . . . . . . . . . . 53
3.6 Vorticity and Divergence Equations . . . . . . . . . . . . . . . . . . . . 60
3.7 Streamfunction and Velocity Potential . . . . . . . . . . . . . . . . . . 66
3.8 The Thermodynamic Equation . . . . . . . . . . . . . . . . . . . . . . . 67
3.9 Summary of PEAK Model Equations . . . . . . . . . . . . . . . . . . . 67
3.10 Conservation of Total Energy . . . . . . . . . . . . . . . . . . . . . . . 69
3.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
vii
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viii Contents
4 The Shallow-Water Model 75
4.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Formulation of theShallow-Water Model . . . . . . . . . . . . . . . . . 78
4.3 Energetics of the SWM . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Angular Momentum in theSWM . . . . . . . . . . . . . . . . . . . . . 83
4.5 Vorticity and Divergence Equations . . . . . . . . . . . . . . . . . . . . 84
4.6 The SWMon theSphere . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.7 Nonlinear Balance of Mass and Wind . . . . . . . . . . . . . . . . . . . 88
4.8 A Stationary Solution on theSphere . . . . . . . . . . . . . . . . . . . 95
4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5 The Barotropic Vorticity Equation 103
5.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 The Nondivergent Barotropic Model. . . . . . . . . . . . . . . . . . . . 104
5.3 The Barotropic Model in Spherical Geometry . . . . . . . . . . . . . . 105
5.4 Rossby–Haurwitz Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.5 Barotropic Instability—By Example . . . . . . . . . . . . . . . . . . . . 110
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Balanced Flow 115
6.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 QG Scaling in theSWM . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Baroclinic QGFlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4 Jet Stream Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.5 HydrodynamicFlow Instability . . . . . . . . . . . . . . . . . . . . . . 145
6.6 Spectral Vertical Normal-Mode Decomposition . . . . . . . . . . . . . . 155
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
II Spectral Numerical Models 167
7 The Spectral Method 169
7.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.2 Series-Expansion Methods . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.3 The Spectral Method in NWP . . . . . . . . . . . . . . . . . . . . . . . 176
7.4 Galerkin and Collocation—Linear Advection . . . . . . . . . . . . . . . 182
7.5 Galerkin and Collocation—Nonlinear Advection . . . . . . . . . . . . . 185
7.6 Galerkin and Collocation—Discussion . . . . . . . . . . . . . . . . . . . 187
7.7 The Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.8 Galerkin and Collocation—Burgers Equation . . . . . . . . . . . . . . . 197
7.9 Aliasing—A Bit of Theory . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.10 Aliasing—The Simplest Example . . . . . . . . . . . . . . . . . . . . . 223
7.11 Spectral Expansion in NWP Models . . . . . . . . . . . . . . . . . . . . 228
7.12 Routinesp2gg and the Collocation Grid. . . . . . . . . . . . . . . . . . 232
7.13 Routinegg2sp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
7.14 The Transform Method in the BVE . . . . . . . . . . . . . . . . . . . . 245
7.15 Spectral Isotropic Correlations . . . . . . . . . . . . . . . . . . . . . . . 250
7.16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
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Contents ix
8 Vertical Discretization 257
8.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8.2 Vertical Staggering of Variables . . . . . . . . . . . . . . . . . . . . . . 258
8.3 Formulation of theVertical Discretization. . . . . . . . . . . . . . . . . 260
8.4 Energetics of theVertical Discretization. . . . . . . . . . . . . . . . . . 269
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
9 Time Integration 275
9.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 275
9.2 Gravity-WaveSeparation . . . . . . . . . . . . . . . . . . . . . . . . . . 276
9.3 Semi-Implicit Time Differencing . . . . . . . . . . . . . . . . . . . . . . 283
9.4 Leapfrog Time Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . 290
9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
10 Code Structure of PEAK 297
10.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 297
10.2 Numerical Implementation of PEAK . . . . . . . . . . . . . . . . . . . . 302
10.3 Description of Standard PEAK Routines . . . . . . . . . . . . . . . . . 309
10.4 Spectral-Transform Routines . . . . . . . . . . . . . . . . . . . . . . . . 338
10.5 Numerical-Recipes-Based Routines . . . . . . . . . . . . . . . . . . . . 339
10.6 Numerical Algorithms Group (NAG)Routines . . . . . . . . . . . . . . 340
10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
11 Experimentation with PEAK 343
11.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 343
11.2 Setting Up PEAK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
11.3 PEAK in NAG-UsingMode (NAGM) . . . . . . . . . . . . . . . . . . . 347
11.4 Validation of PEAK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
12 Barotropic PEAK Configurations 371
12.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 371
12.2 The Spectral BVE Model in PEAK . . . . . . . . . . . . . . . . . . . . 372
12.3 The Spectral SWM in PEAK . . . . . . . . . . . . . . . . . . . . . . . 381
12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
III Appendices 401
A Tensor Analysis 403
A.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 403
A.2 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
A.3 Orthogonal Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . 405
A.4 Standard Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . 411
A.5 Modified Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . 413
A.6 Gradient of a Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
A.7 The Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
A.8 The Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
A.9 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
A.10 The Physical Components of a Vector . . . . . . . . . . . . . . . . . . . 418
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x Contents
A.11 Advection of a Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
A.12 The Helmholtz Decomposition . . . . . . . . . . . . . . . . . . . . . . . 420
A.13 Generalized Mass Continuity . . . . . . . . . . . . . . . . . . . . . . . . 421
A.14 Generalized Vertical Velocity . . . . . . . . . . . . . . . . . . . . . . . . 423
A.15 The HPE Material Derivative . . . . . . . . . . . . . . . . . . . . . . . 424
A.16 Solutions to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . 425
B Spectral Basis Functions 433
B.1 Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . 433
B.2 Tabulated Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . 435
B.3 Computation of Pˆm(µ) in Routine plgndr2 . . . . . . . . . . . . . . . . 435
n
B.4 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
B.5 Gaussian Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
C The PEAK Model Code 441
C.1 The PEAK Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
C.2 Spectral Transform Routines . . . . . . . . . . . . . . . . . . . . . . . . 463
C.3 Numerical-Recipes-Based Routines . . . . . . . . . . . . . . . . . . . . 464
C.4 Spectral Transform Routineswith NAG. . . . . . . . . . . . . . . . . . 466
C.5 Normal-Mode Computations Code . . . . . . . . . . . . . . . . . . . . . 468
C.6 Burgers-Equation Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
C.7 The Fourier-Matrix Code . . . . . . . . . . . . . . . . . . . . . . . . . . 471
Afterword 473
Bibliography 475
Index 493
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Preface
Here, on the level sand, between the sea and land, what
shall I build or write, against the fall of night?
—[Har40, Sec. 7]
This book is about numericalweather prediction (NWP).Itfocuses on theapplica-
tion of the spectral method in NWP models. The spectral method in NWP is described
by (i) illustrating the spectral theory in general terms, as well as by (ii) applying this
theory to the implementation of a specific spectral NWP model. The description of the
implementation ofthismodel, namedPEAK,1 isgivenindetail,soastoilluminateallthe
building blocks that constitute a spectral global NWPmodel.
Thespectraltheorylaidoutherealong(i)isthetopicofseveralotherbooksonNWP.
However,thedetaileddescriptionofapplyingthistheoryinapracticalcontext,along(ii),
asitleadsfromthegenerallyformulated modelequationstotheirfinalimplementationin
terms of the Fortran 90/95 statements of a global NWP model cannot easily be found
elsewhere in onesingle place.
This book is thus unique as it illustrates spectral theory in terms of its practical
applicationanditcontainsallaspectsofaglobalNWPmodelbuiltonthespectralmethod.
Containingsuchacomprehensivemodeldescription,thebook’suniquenesscomesfromits
bringingtogetherinoneplaceallthematerialformingtheframeworkuponwhichweather
forecasting models are built. Due to the complexity of today’s advanced forecast models,
such material documenting their full details is frequently not available in a single place
in a concise, unified, and consistent format. All this material is found in this book and
describedinpracticaltermsinthecontextofPEAKstartingwiththegoverningcontinuous
model equations and endingwith thecoded model.
Essentially, the book contains the algorithmic formulation of the spectral method
whenapplied tothelarge-scale, highly relevantmodelingproblem ofnumerically simulat-
ing atmospheric flows. Thus, if you are interested in physics, engineering, mathematics,
and numerical methods, as well as, in particular, the intriguing subjects of atmospheric
modeling and meteorology, this book brings together for you, in an accessible and inter-
nally consistent form, in oneplace, thephysicalfundamentals that enteran NWPmodel,
thegeneral theoryofspectralmethods,as wellasthedemonstration of howboth ofthese
cornerstones end up in the Fortran 90/95 code of a global spectral model.
As such, the book serves as a reference and a stepping stone for delving into the
complexities of the highly advanced models as they are nowadays operationally used for
1The abbreviation PEAK stands forPrimitive-Equation Atmospheric Research Model Kernel.
Table1.1contains alistofotherabbreviations used.
xi
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xii Preface
weatherandclimateforecasting. Bringingtogetherthespectralandphysicalfundamentals
entering NWP models, and thus providing for such a stepping stone, fills an important
gap in the literature on atmospheric numerical modeling. The desire to fill this gap in a
comprehensive and consistent fashion has been the strongest motivation for me to create
this book. Many times have I, for example, converted between different sign conventions
and normalizations used for spherical harmonics in different models, or worked out the
metrictermsthatappearin theadvectivetermsin thevorticity equationtoseehowthey
may havespecifically been formulated in various applications.
Theaimofthisbookistoprovidefullinsightintotheuseofthespectralmethodin
NWP models. This insight is made specific in terms of demonstrating its application in
buildingtheNWPmodelPEAK.Inthisapplication,thetransformmethodoccupies,asin
other spectral NWP models, a central place. Ever since it had been proposed in 1970 as
anefficientmethodforthecomputationofnonlineartermsinspectralmodels,ithasbeen
most instrumental in making spectral methods famous, widely popular, and practically
applicable to numerous modeling problems. The computational advantages achieved by
using the transform method that are inherently linked to the availability of the discrete
fast Fourier transform (FFT) have moved the spectral method to the most widely used
numerical techniquewhen addressing global NWPproblems.
By working through this book, you can hope to gain a self-contained and concise
overviewofimportantfundamentalsofatmosphericdynamicsandthespectralmethodas
usedinNWP,aswellasafullillustration ofthedetailsthatarerequiredforconstructing
a spectral NWP model. The construction of such a model, exemplified here in terms of
PEAK, may be viewed as a four-stage process, consisting of (i) the formulation of the
continuous model equations, followed by (ii) the discretization of these model equations,
(iii) the coding of the discretized model equations in a high-level computer programming
language, and finally (iv) the validation of the numerical model. All of these four steps
are discussed comprehensively here.
My main intentionin writing thisbook hasbeentobringtogether inoneplace and
inaconsistent format, all thematerial requiredfor constructingthedynamicalcoreof an
atmospheric spectral NWP model, building on the general theory of spectral numerical
modeling and of atmospheric dynamics. Providing such a comprehensive and internally
consistent reference on the details of a spectral dynamical core is important, since it is
such dynamical cores around which further components of state-of-the-art NWP models
are built. After all, these state-of-the-art models are the basis for producing the weather
forecastswehavebecomesousedtohavingavailableontelevision,radio,andtheinternet.
The idea of making tangible, by writing down in this book, the details that are
behindtheseforecastsdidbecomearealitywhenthemodelvalidationproposalby[PST04]
appeared. The validation procedure described in this paper as being reproducible by
any NWP model offered a stringent benchmark when carrying out for PEAK the above-
mentionedvalidationstep(iv),andassuchallowedforcompletingtheformulation-develop-
ment–validation chain prototypical for any model design process.
Asyouwillfindout,Iamnotagreatbookwriter. However,Ihaveundergoneevery
effort to present the material as comprehensively and as clearly as possible, by writing
down many intermediate steps when going from one result to the next that might have
been omitted otherwise. At the same time, I carefully refer to places where you might
find even more detailed information, by including a specific pointer when referencing a
publication. ThroughthiseffortIhavetriedtoavoidwritingdownstatementsorequations
that cannot be understood by themselves without offering a hint of where to research for
furtherinformation.
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